1000/1000

Hot
Most Recent

Submitted Successfully!

To reward your contribution, here is a gift for you: A free trial for our video production service.

Thank you for your contribution! You can also upload a video entry or images related to this topic.

Do you have a full video?

Are you sure to Delete?

Cite

If you have any further questions, please contact Encyclopedia Editorial Office.

Carmona, J.M.; Cortés, J.L.; Relancio, J.J.; Reyes, M.A. Beyond Special Relativity. Encyclopedia. Available online: https://encyclopedia.pub/entry/25064 (accessed on 04 November 2024).

Carmona JM, Cortés JL, Relancio JJ, Reyes MA. Beyond Special Relativity. Encyclopedia. Available at: https://encyclopedia.pub/entry/25064. Accessed November 04, 2024.

Carmona, José Manuel, José Luis Cortés, José Javier Relancio, Maykoll A. Reyes. "Beyond Special Relativity" *Encyclopedia*, https://encyclopedia.pub/entry/25064 (accessed November 04, 2024).

Carmona, J.M., Cortés, J.L., Relancio, J.J., & Reyes, M.A. (2022, July 12). Beyond Special Relativity. In *Encyclopedia*. https://encyclopedia.pub/entry/25064

Carmona, José Manuel, et al. "Beyond Special Relativity." *Encyclopedia*. Web. 12 July, 2022.

Copy Citation

There are two different ways in which one can go beyond the kinematics of Special Relativity (SR). One can consider adding to the Standard Model (SM) Lagrangian new terms that violate Lorentz Invariance (LIV). In case one wants to preserve the relativistic invariance, one should modify the transformations between inertial frames and accordingly modify the special relativistic kinematics; this is what is called Doubly/Deformed Special Relativity (DSR).

Quantum Gravity Phenomenology
Doubly Special Relativity
Lorentz Invariance Violation

A deviation from SR whose effects increase with the energy can be incorporated in the framework of Effective Field Theory. This is achieved by adding to the fields and symmetries that define the SM of particle physics, terms of dimension higher than four which are not invariant under boosts (neglecting a possible deviation in the rotational symmetry). This is known as the Standard Model Extension (SME) ^{[1]}^{[2]}.

The most important effect of this extension is contained in the free part of the Lagrangian density, i.e., in the part which is quadratic in fields. This leads to a modification of the SR energy–momentum relation of a free particle (modified dispersion relation)

$$E\phantom{\rule{0.166667em}{0ex}}\approx \phantom{\rule{0.166667em}{0ex}}p+\frac{{m}^{2}}{2\phantom{\rule{0.166667em}{0ex}}p}+\alpha \phantom{\rule{0.166667em}{0ex}}\frac{{p}^{n+1}}{{\mathsf{\Lambda}}^{n}}\phantom{\rule{1.em}{0ex}}\mathrm{when}\phantom{\rule{1.em}{0ex}}m\ll p\ll \mathsf{\Lambda}\phantom{\rule{0.166667em}{0ex}},$$

where E and p are the energy and the modulus of the momentum of a particle with mass m, respectively, Λ and n are the energy scale and order of correction which parametrize the deviations from SR, respectively, and α is a dimensionless constant which parametrizes the particle’s dependence of the LIV effects.

The dimension of the first quadratic term in the SME which violates Lorentz invariance is D=(4+n). There are two alternative values, n=1 (linear case) or n=2 (quadratic case), considered in the studies of LIV. It is also possible to consider a minimal SME ^{[3]}, where there are only operators of dimension four or less.

A modified dispersion relation implies a modification of the expression of the velocity of a particle in terms of the energy, which can lead to observable consequences from transient astrophysical phenomena (energy-dependent photon time delays), even if the energies of the observed particles are much smaller than the energy scale parametrizing the LIV. Another observable consequence of modified dispersion relations is the modification of the SR kinematics in the different particle processes which are relevant in high-energy astrophysics. The thresholds and the separation of kinematically allowed/forbidden processes (with respect to SR) are affected by the modified energy–momentum relation when the mass-dependent and the LIV terms in (1) become comparable, i.e., when (m^{2}/E^{2})∼(E/Λ)^{n} ^{[4]}^{[5]}^{[6]}. This happens for E≪Λ, and then, one can have observable consequences of the LIV in high-energy astrophysics at energies much lower than the energy scale of LIV.

It was seen that considering an LIV scenario entails a loss of the relativity principle and the acceptance of a preferred reference frame, which is usually identified with the one defined by the homogeneity and isotropy of the Cosmic Microwave Background (CMB). If one wants to maintain a relativity principle when going beyond SR, one has to consider a deformation of the transformations relating the inertial reference frames. The deformation of SR, usually called DSR ^{[7]}, is assumed to be parametrized by a new energy scale Λ, which does not usually affect the rotational symmetry, as in the case of LIV.

A necessary ingredient of this departure from SR at the kinematical level is a nontrivial characterization of a multi-particle system with a total energy and momentum differing from the sum of the energies and momenta of the particles. One then has a composition of energy and momentum which is non-symmetric under the exchange of the particles ^{[8]}. One arrives at this conclusion from different perspectives of DSR. The starting point of this proposal is the attempt to make compatible the relativistic invariance with the presence of a minimal length ^{[9]}^{[10]}, which seems to be a characteristic of a quantum theory which incorporates consistently the gravitational interaction ^{[11]}^{[12]}^{[13]}^{[14]}. Such minimal length can be understood as a consequence of a non-commutativity in a generalization of the classical spacetime, which requires us to go beyond the usual implementation of continuous symmetries by Lie algebras. The new algebraic structure is a Hopf algebra ^{[15]} with a non-trivial co-product, which leads to a deformed kinematics with a non-symmetric composition of momenta ^{[16]}^{[17]}. An alternative way to arrive at the same conclusion is to identify the non-commutativity of spacetime with a non-commutativity of translations in a curved momentum space, which can also be related with to composition of momenta ^{[8]}^{[18]}^{[19]}^{[20]}^{[21]}. This composition law is therefore a crucial ingredient differentiating DSR and LIV.

Together with the non-linear composition of momenta, the invariance under deformed Lorentz transformations will lead in many cases to a modification of the dispersion relation. As a consequence, in the kinematic analysis of a process in DSR, one has to consider both a possible modification of the energy–momentum relation of the particles participating in it and a modification of the energy–momentum conservation law. The compatibility with the relativity principle, in comparison with the case of LIV, can be shown to produce a cancellation of the effects of the two modifications. Therefore, in order to have an observable consequence of the deformation of the kinematics in a process, one has to consider energies comparable to the energy scale Λ of the deformation ^{[22]}^{[23]}^{[24]}^{[25]}^{[26]}. This means that in order to have a signal of DSR in the particle processes which are relevant in high-energy astrophysics, it is necessary to consider an energy scale parametrizing the deformation of the kinematics of the order of the energy involved in those processes. At the same time, many of the constraints to the high-energy scale in the case of LIV do not apply in the DSR scenario.

The two previous kinematic ingredients of DSR raise several problems and apparent contradictions on the physical interpretation of the theory. On the one hand, a modification of the composition of momenta in a particle system (independently of the distance between the particles) implies a departure from the notion of absolute locality in spacetime ^{[27]}^{[28]}. The corresponding loss of the crucial property of cluster ^{[29]}, which is at the basis of the formulation of special-relativistic quantum field theory, originates the so-called spectator problem ^{[30]}^{[31]}^{[32]}. On the other hand, a modification of the dispersion relation with the associated modification of the velocity of a particle raises an apparent inconsistency of DSR when one applies the deformed kinematics to any system, including a macroscopic system (soccer ball problem ^{[33]}^{[34]}).

- Colladay, D.; Kostelecky, V.A. Lorentz violating extension of the standard model. Phys. Rev. 1998, D58, 116002.
- Kostelecky, V.A.; Russell, N. Data Tables for Lorentz and CPT Violation. Rev. Mod. Phys. 2011, 83, 11–31.
- Colladay, D.; Kostelecky, V.A. CPT violation and the standard model. Phys. Rev. D 1997, 55, 6760–6774.
- Addazi, A.; Alvarez-Muniz, J.; Batista, R.A.; Amelino-Camelia, G.; Antonelli, V.; Arzano, M.; Asorey, M.; Atteia, J.-L.; Bahamonde, S.; Bajardi, F.; et al. Quantum gravity phenomenology at the dawn of the multi-messenger era—A review. Prog. Part. Nucl. Phys. 2022, 125, 103948.
- Mattingly, D. Modern tests of Lorentz invariance. Living Rev.Rel. 2005, 8, 5.
- Liberati, S. Tests of Lorentz invariance: A 2013 update. Class. Quant. Grav. 2013, 30, 133001.
- Amelino-Camelia, G. Quantum-Spacetime Phenomenology. Living Rev. Rel. 2013, 16, 5.
- Kowalski-Glikman, J. De sitter space as an arena for doubly special relativity. Phys. Lett. B 2002, 547, 291–296.
- Amelino-Camelia, G. Testable scenario for relativity with minimum length. Phys. Lett. B 2001, 510, 255–263.
- Amelino-Camelia, G. Relativity in space-times with short distance structure governed by an observer independent (Planckian) length scale. Int. J. Mod. Phys. D 2002, 11, 35–60.
- Kato, M. Particle Theories With Minimum Observable Length and Open String Theory. Phys. Lett. 1990, B245, 43–47.
- Susskind, L. String theory and the principles of black hole complementarity. Phys. Rev. Lett. 1993, 71, 2367–2368.
- Garay, L.J. Quantum gravity and minimum length. Int. J. Mod. Phys. 1995, A10, 145–166.
- Hossenfelder, S. Minimal Length Scale Scenarios for Quantum Gravity. Living Rev.Rel. 2013, 16, 2.
- Majid, S. Foundations of Quantum Group Theory; Cambridge University Press: Cambridge, UK, 1995.
- Majid, S.; Ruegg, H. Bicrossproduct structure of kappa Poincare group and noncommutative geometry. Phys. Lett. 1994, B334, 348–354.
- Lukierski, J.; Nowicki, A.; Ruegg, H. New quantum Poincare algebra and k deformed field theory. Phys. Lett. 1992, B293, 344–352.
- Carmona, J.M.; Cortés, J.L.; Relancio, J.J. Relativistic deformed kinematics from momentum space geometry. Phys. Rev. 2019, D100, 104031.
- Lizzi, F.; Manfredonia, M.; Mercati, F. The momentum spaces of κ-Minkowski noncommutative spacetime. Nucl. Phys. B 2020, 958, 115117.
- Carmona, J.M.; Cortés, J.L.; Relancio, J.J. Curved Momentum Space, Locality, and Generalized Space-Time. Universe 2021, 7, 99.
- Relancio, J.J. Geometry of multiparticle systems with a relativistic deformed kinematics and the relative locality principle. Phys. Rev. D 2021, 104, 024017.
- Carmona, J.M.; Cortes, J.L.; Romeo, B. Nonuniversal relativistic kinematics. Phys. Rev. D 2015, 91, 085036.
- Albalate, G.; Carmona, J.M.; Cortés, J.L.; Relancio, J.J. Twin Peaks: A possible signal in the production of resonances beyond special relativity. Symmetry 2018, 10, 432.
- Relancio, J.; Liberati, S. Constraints on the deformation scale of a geometry in the cotangent bundle. Phys. Rev. D 2020, 102, 104025.
- Carmona, J.M.; Cortés, J.L.; Pereira, L.; Relancio, J.J. Bounds on Relativistic Deformed Kinematics from the Physics of the Universe Transparency. Symmetry 2020, 12, 1298.
- Carmona, J.M.; Cortés, J.L.; Relancio, J.J.; Reyes, M.A.; Vincueria, A. Modification of the mean free path of very high energy photons due to a relativistic deformed kinematics. arXiv 2021, arXiv:2109.08402.
- Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. The principle of relative locality. Phys. Rev. 2011, D84, 084010.
- Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. Relative locality: A deepening of the relativity principle. Gen. Rel. Grav. 2011, 43, 2547–2553.
- Carmona, J.M.; Cortes, J.L.; Relancio, J.J. Spacetime and deformations of special relativistic kinematics. Symmetry 2019, 11, 1401.
- Carmona, J.M.; Cortes, J.L.; Mazon, D.; Mercati, F. About Locality and the Relativity Principle Beyond Special Relativity. Phys. Rev. D 2011, 84, 085010.
- Amelino-Camelia, G. On the fate of Lorentz symmetry in relative-locality momentum spaces. Phys. Rev. D 2012, 85, 084034.
- Gubitosi, G.; Heefer, S. Relativistic compatibility of the interacting κ-Poincaré model and implications for the relative locality framework. Phys. Rev. D 2019, 99, 086019.
- Hossenfelder, S. Multi-Particle States in Deformed Special Relativity. Phys.Rev. 2007, D75, 105005.
- Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. Relative locality and the soccer ball problem. Phys. Rev. D 2011, 84, 087702.

More

Information

Subjects:
Physics, Particles & Fields

Contributors
MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register
:

View Times:
520

Update Date:
13 Jul 2022